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{
}
=
tion precision at any krigged location. If the nugget
variance is large, it may produce undesirable large
estimation variances (Castrignanò, 2008).
Fz
(; ( )
u
n
=
Prob
Z
( )
u
³
zn
( )
k
1,2,...K
k
k
(9)
A non-parametric geostatistical estimation of
ccdf values is based on the interpretation of the
conditional probability (eq. 9) as the conditional
expectation of an indicator random variable I (u;
z
k
) given the information (n) (Van Meirvenne &
Goovaerts, 2001):
Modeling of Local Uncertainty
The value of the attribute z at an unsampled loca-
tion u is certainly affected by uncertainty.
The common geostatistical approach to
model local uncertainty implies the calculation
of a kriging estimate and the associated error
variance. A more rigorous approach is to model
the uncertainty about the unknown z(u) before
and independently of the choice of a particular
estimate for that unknown (Goovaerts, 1999).
This means modeling the uncertainty about the z
value at location u through a random variable Z(u)
that is characterized by its distribution function
(Goovaerts, 1999):
=
{
}
FzNEI
(; ( )
u
( ;
u
zn
()
k
k
(10)
with I (u; z
k
) =1 if Z(u)≥ z
k
and zero otherwise,
can be considered as the conditional probability
in Eq.5. Ccdf values can thus be estimated by
means of ordinary kriging of indicator transforms
of data.
Indicator Coding in Presence of Soft Data
{
}
Fzn
(; ())
u
=
Prob
Z
( )
u
³
zn
( )
(8)
The indicator approach requires a preliminary
coding of each observation z(u
α
) into a series
of K values indicating whether the threshold z
k
is exceeded a or not. If the measurement errors
are assumed negligible compared to the spatial
variability, observations are coded into hard (0 or
1) indicator data (Van Meirvenne & Goovaerts,
2001):
where the notation “|(n)” expresses the condi-
tioning to the n data z(u
α
). The function F(u;z|(n))
is called conditional cumulative distribution func-
tion (ccdf). The ccdf fully models the uncertainty
at location u since it gives the probability that
the unknown is greater than any given threshold
z (Van Meirvenne & Goovaerts, 2001). In order
to determine a ccdf it is possible to make use of
parametric or non - parametric approaches.
In a parametric approach, the determination
of the ccdf is straightforward by means of the
assignation of an analytical model that results
completely defined by some parameters charac-
teristic of the type of distribution. On the contrary,
the non parametric algorithm is not based on an
a-priori assumption about the form of the analyti-
cal expression of F(u;z|(n)), therefore it is more
flexible. It consists of estimating the value of the
ccdf for a series of K threshold values z
k
, discretiz-
ing the range of variation of z (Van Meirvenne &
Goovaerts, 2001):
ì
í
ï
ï
î
ï
ï
1 if z(u)
³
z
a
k
i(u;z)
=
k
=
1,2,...K
a
k
0 otherwise
(11)
The major advantage of the indicator approach
is its ability to incorporate in the algorithm of
calculation different types of information: “soft”
information, that is to say qualitative information,
less precise, e.g., soil map or qualitative field
observations such as the smell or color of con-
taminated soil in addition to direct measurements
on the attribute of interest, that corresponds to the
“hard” information. The only requirement is that
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